MathNet 2024-part-04 — 168 problems (EN–VI) Question 11

A teacher wants to spend a morning practising Olympiad problems with her class in teams. To this end, she has set up an arrangement with six large tables. At each table several students can work together on a problem. Each table has a problem; there are three different problems, each on two tables. There are three rounds in which each student sits at one of the tables. The teacher wants to schedule the students in such a way that after three rounds, each student has worked on all the problems once. In addition, she wants each pair of students to sit at the same table at most once.
What is the maximum number of students for which such a schedule is possible?